Basically, it uses operator method to solve linear non-homogeneous ordinary differential equation with variable coefficients. If you have a differential operator of order n, and you know a fundamental solution to the operator, you can use my method to factorize a linear factor out of the differential operator. By repeating the process, you can factorize the operator totally, and solving the equation is a matter of repeated integration only. (It doesn't teach you how to find a fundamental solution though. It is done probably by guessing.)

I am not sure if this method is original. I just think of it a few days ago. People may have used it for a long time. But I can't find anything similar from the web.

I didn't go through the detailed algebra, but to me it smells like "reduction of order". ross_tang, you should look this up in any ode book to see if your approach is really any different from the standard approach.

Of course, Ben Niehoff is right that the "if we can find a fundamental solution to the differential operator ..." step in the method is the really hard part. One might hope that once you guess a solution to the nth order equation that the resulting n-1 order equation would be even easier to guess a solution to, but I expect there are counterexamples to that hope.

So basically, your claim is that if you already know all the solutions, then you can find the solutions...

It is not you know all solutions. You know one solution of the homogeneous equation of order n, and can reduce the equation to n-1 order. And now, you just need to find a solution to the n-1 order equation. It may be simpler, but it may be more difficult too.

@jasonRF,
You may say my method is something like "reduction of order", but it is different. In reduction of order, you are finding all fundamental solution to the original homogeneous equation. After you do that, you need to use method of variation of parameters to obtain the particular solution.

In my method, I am reducing the differential operator into products of linear factors. Once you have done that, you can find the particular solution and all fundamental solutions to the non-homogeneous equation at once by applying the formula of integrating factor.

Finally, it is true that the most difficult part is finding any particular solution. I just want to give an alternative to method of variation of parameters and reduction of order, into factorization of operator. I think it is a much more neat method.

In short, my method is just like long division in factorizing polynomial of degree n. Once you know a root of the polynomial equation, you can use long division to reduce it to a linear factor multiply by a polynomial of degree n-1. It is proved that we can't have analytic formula for degree n>=5. Therefore, if we can guess one of the root, we can reduce its degree. But for the case of linear differential equation, we resort to guessing even for 2nd order only.